Generalizations of Chung-Feller Theorem

TL;DR

This paper generalizes Chung-Feller theorems to weighted lattice paths, establishing connections between Dyck and Motzkin path properties and revealing that many paths exhibit both types of Chung-Feller properties.

Contribution

It introduces generalized Chung-Feller theorems for weighted lattice paths, linking Dyck and Motzkin path properties and expanding the scope of classical combinatorial results.

Findings

Proved Chung-Feller theorems of Dyck type for three classes of paths.

Established Chung-Feller theorems of Motzkin type for two classes.

Discovered many paths possess both Dyck and Motzkin type properties.

Abstract

The classical Chung-Feller theorem [2] tells us that the number of Dyck paths of length $n$ with flaws $m$ is the $n$-th Catalan number and independent on $m$. L. Shapiro [7] found the Chung-Feller properties for the Motzkin paths. In this paper, we find the connections between these two Chung-Feller theorems. We focus on the weighted versions of three classes of lattice paths and give the generalizations of the above two theorems. We prove the Chung-Feller theorems of Dyck type for these three classes of lattice paths and the Chung-Feller theorems of Motzkin type for two of these three classes. From the obtained results, we find an interesting fact that many lattice paths have the Chung-Feller properties of both Dyck type and Motzkin type.